Uncertainty Quantification with MLMC
This section will be dedicated to demonstrating how Flooddrake can be used with Firedrake-mlmc (available at https://github.com/firedrakeproject/firedrake-mlmc) to solve uncertainty quantification shallow water problems using multilevel Monte Carlo.
An uncertain flooding problem will be consider. The initial profile of the free-surface depth is shown below, with the topography of the land given by a white mesh, and flooded land shown in blue. The darker areas of the topography represent dips and bumps. The sides of the topography are tilted to create river banks. This can be seen by the 3D profile also below.
The state, with a log-normally distributed magnitude of rainfall (spatially uniform), is simulated over the time interval [0, 500]. Water flows down the bumpy river banks to flood the bottom of the river and create pools of water in the dips.
A standard Monte Carlo simulation of this model could be carried out to approximate statistics of the random field solution to this problem. However, given the cubic rate of complexity cost growth with the increase in resolution of this problem (see below scaling plot), one finds that approximating these statistics with a high accuracy (low discretization bias / low variance) we would require an infeasible amount of computational effort.
Thus we consider the use of the multilevel Monte Carlo method to reduce the computational complexity of this uncertainty quantification. This uses a hierarchy of ensembles containing realisations of the flooding system using different spatio-temporal resolutions. We use a hierarchy of 4 levels of resolution, with grid sizes of 12x12, 24x24, 48x48 and 96x96 cells. Suppose we desire the mean free-surface depth field of this system at the end of the time interval. Using Firedrake-mlmc (documentation on how to carry out uncertainty quantification problems can be seen at https://github.com/firedrakeproject/firedrake-mlmc) one can investigate the computational cost benefits of estimating this quantity with a multilevel Monte Carlo approximation of a given accuracy with respect to a standard Monte Carlo approximation.
The below figure shows the speed-up of estimating the mean field of this flooding system with increasing accuracy (norm of the Root-Mean-Square-Error away from an extremely accurate gold-standard Monte Carlo estimate) against the standard Monte Carlo estimator. For increasing accuracy, past that shown on this figure, when the computational cost scaling is at it's highest rate (past some overheads shown for low resolutions in the previous figure) we should expect to see greater speed-ups.
